Theorem fvopab4ndm 6973

Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)

Hypothesis

Ref	Expression
fvopab4ndm.1	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

Assertion

Ref	Expression
fvopab4ndm	⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅)

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvopab4ndm

Step	Hyp	Ref	Expression
1		fvopab4ndm.1	. . . . 5 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2	1	dmeqi 5853	. . . 4 ⊢ dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3		dmopabss 5867	. . . 4 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
4	2, 3	eqsstri 3968	. . 3 ⊢ dom 𝐹 ⊆ 𝐴
5	4	sseli 3918	. 2 ⊢ (𝐵 ∈ dom 𝐹 → 𝐵 ∈ 𝐴)
6		ndmfv 6866	. 2 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
7	5, 6	nsyl5 159	1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∅c0 4268  {copab 5141  dom cdm 5625  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-dm 5635  df-iota 6448  df-fv 6500

This theorem is referenced by:  fvmptndm  6974